Mathematical model of vortex penetration phenomenon

Vortex penetration affects the stability of a superconducting system and limits the possible application of the system. However, the mathematical description to this phenomenon is currently unavailable. Here I present a mathematical model in which I consider the effects of bulk pinning and internal field repulsive force on vortex hopping. Thereafter, I proposed a series expansion to the activation energy and derived a general formula for describing the time dependence of the vortex penetration process. With these formulas, I can analyze the experimental data and calculate the activation energy of the vortex penetration phenomenon. The results are accurate for the time dependence of the internal field measurements in a $Bi_2Sr_2CaCu_2O_{8+x}$ superconductor.